Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization

Abstract

In topological data analysis and visualization, topological descriptors such as persistence diagrams, merge trees, contour trees, Reeb graphs, and Morse-Smale complexes play an essential role in capturing the shape of scalar field data. We present a state-of-the-art report on scalar field comparison using topological descriptors. We provide a taxonomy of existing approaches based on visualization tasks associated with three categories of data: single fields, time-varying fields, and ensembles. These tasks include symmetry detection, periodicity detection, key event/feature detection, feature tracking, clustering, and structure statistics. Our main contributions include the formulation of a set of desirable mathematical and computational properties of comparative measures, and the classification of visualization tasks and applications that are enabled by these measures.

Realizable piecewise linear paths of persistence diagrams with Reeb graphs

“…Reeb graphs have become an important tool in topological data analysis for the purpose of visualizing continuous functions on complex spaces, as they yield a simplified discrete structure. Originally developed in relation to Morse theory [29], these objects are used extensively for shape comparison, constructing skeletons of data sets, surface simplification, and visualization; for more details on these and more applications, we refer to recent surveys on the topic [5,31]. While some information is lost in the construction of the Reeb graph, such simplified structures allow for more efficient methods to analyze and compare data sets.…”